OBJ files coming out gray, no color
Depends in what way the mesh is colored. Vertex colors or texture. Obj file format is officially not able to transfer Vertex color. You would need another file format to do so. FBX for example.

When the "color" is a texture, then this texture is most probably missing. Have a look that the texture is in the same folder than the obj file, and that there is a *.mtl file available. That's part of the *.obj file, and defines the material settings. 

When this doesn't fix it then the path to the texture inside the *.mtl file might be wrong. Most probably an absolute path with a C:\ at the beginning. Like C:\myfolder\mytexture.png. Open it with a text editor, search for the map_kd part and change it to a relative path. Which is basically just the texture name: mytexture.png

Cycles is not able to import the texture from an obj file yet. You have to have Blender Internal renderer activated. So when the previous step also doesn't help, then simply add the texture inside Blender.